Gabriel Birzu - Hidden Phase Transitions in Reaction-Diffusion Waves

Traveling waves in reaction-diffusion models describe a wide range of phenomena: from combustion in physics to invasion of foreign species in population biology. All such waves can be classified as either pulled or pushed depending on how the growth rate depends on population density. The propagation of pulled waves is controlled by the fast reproduction at the leading edge, while pushed waves advance via the “spill-over” of the growth in the bulk. This classification captures major differences in the properties of pulled and pushed waves including the expansion velocity and the shape of the expansion profile. However, the internal dynamics inside the wave, such as the loss of genetic diversity, are also important in many applications. We find that the rate of diversity loss divides pushed waves into two distinct subclasses. For strongly pushed waves, the lifetime of a genetic polymorphism is proportional to the population density. In contrast for weakly pushed waves, this dependence is a sub-linear power law, with the exponent controlled solely by the distance to the universality class of pulled waves. Other aspects of internal dynamics further demonstrate that not all pushed waves are the same. Our results are a striking example of how ecological properties of the species such the density-dependence of its growth modulate the course of evolution by altering the amount of genetic diversity. The differences in the adaptive potential of the two subclasses of pushed waves have important implications for pest management, control of invasive species, and cancer progression.